It is simply two different ways of defining the integration domain, which is shown in figure 12.28.

Since both parameters can be 0 to ##\infty##, the outer integral necessarily needs to be from 0 to ##\infty##. The inner integral must be such that you integrate over the shaded domain and nothing else. If you have fixed ##t## what values can ##\tau## take within this domain? If you have fixed ##\tau##, what values can ##t## take in this domain?

That's right, the visual picture on the 2nd page really helps to see this.
The original inside integral (which you have posted 2nd marskman) is integrating first from left to right or, from t= τ (tau) to ∞, then the outside covers the bounds (τ=0 to τ=∞) But then by reversing the order you'll go from τ=0 to τ=t as the new inside function then cover the remaining bounds, again, from t=0 to t=∞.

Manipulating the order of integration made many problems easier in multivariable calc.